{"id":7937,"date":"2022-09-11T01:43:18","date_gmt":"2022-09-10T20:13:18","guid":{"rendered":"https:\/\/jassweb.com\/solved\/solved-given-1-a-10-1-%e2%89%a4-n-%e2%89%a4-100000-show-how-to-compute-the-value-of-1-x-a-2-x-a2-3-x-a3-n-x-an-efficiently-i-e-in-olog-n\/"},"modified":"2022-09-11T01:43:18","modified_gmt":"2022-09-10T20:13:18","slug":"solved-given-1-a-10-1-%e2%89%a4-n-%e2%89%a4-100000-show-how-to-compute-the-value-of-1-x-a-2-x-a2-3-x-a3-n-x-an-efficiently-i-e-in-olog-n","status":"publish","type":"post","link":"https:\/\/jassweb.com\/solved\/solved-given-1-a-10-1-%e2%89%a4-n-%e2%89%a4-100000-show-how-to-compute-the-value-of-1-x-a-2-x-a2-3-x-a3-n-x-an-efficiently-i-e-in-olog-n\/","title":{"rendered":"[Solved] Given 1 < a < 10, 1 \u2264 n \u2264 100000, show how to compute the value of 1 \u00d7 a + 2 \u00d7 a^2 + 3 \u00d7 a^3 + . . . + n \u00d7 a^n efficiently, i.e. in O(log n)!"},"content":{"rendered":"<p> [ad_1]<br \/>\n<\/p>\n<div id=\"answer-36695996\" class=\"answer js-answer accepted-answer js-accepted-answer\" data-answerid=\"36695996\" data-parentid=\"36693743\" data-score=\"2\" data-position-on-page=\"2\" data-highest-scored=\"0\" data-question-has-accepted-highest-score=\"0\" itemprop=\"suggestedAnswer\" itemscope itemtype=\"https:\/\/schema.org\/Answer\">\n<div class=\"post-layout\">\n<div class=\"votecell post-layout--left\"><\/div>\n<div class=\"answercell post-layout--right\">\n<div class=\"s-prose js-post-body\" itemprop=\"text\">\n<p>If this is homework I guess your professor explained you how to compute Fibonnacci numbers and expect you to use the same trick.<\/p>\n<p>So basically remark that:<\/p>\n<pre><code>|1+a+ a\u00b2+...+ a^n|   |a 0 1| |1+a+ a\u00b2+...+     a^(n-1)|\n|  a+2a\u00b2+...+na^n| = |a a 1| |  a+2a\u00b2+...+(n-1)a^(n-1)|\n|1               |   |0 0 1| |1                       |\n\n                     |a 0 1|^n  |1|\n                   = |a a 1|    |0|\n                     |0 0 1|    |1|\n<\/code><\/pre>\n<p>You need to do a matrix exponentiation, which is <code>O(log(n))<\/code> if we don&#8217;t take into account the precision and only use standard machine&#8217;s operation.<\/p>\n<\/p><\/div>\n<div class=\"mt24\"><\/div>\n<\/div>\n<p>            <span class=\"d-none\" itemprop=\"commentCount\"><\/span> <\/p><\/div>\n<\/div>\n<p>[ad_2]<\/p>\n<p>solved Given 1 < a < 10, 1 \u2264 n \u2264 100000, show how to compute the value of 1 \u00d7 a + 2 \u00d7 a^2 + 3 \u00d7 a^3 + . . . + n \u00d7 a^n efficiently, i.e. in O(log n)! \n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>[ad_1] If this is homework I guess your professor explained you how to compute Fibonnacci numbers and expect you to use the same trick. So basically remark that: |1+a+ a\u00b2+&#8230;+ a^n| |a 0 1| |1+a+ a\u00b2+&#8230;+ a^(n-1)| | a+2a\u00b2+&#8230;+na^n| = |a a 1| | a+2a\u00b2+&#8230;+(n-1)a^(n-1)| |1 | |0 0 1| |1 | |a 0 1|^n &#8230; <a title=\"[Solved] Given 1 &lt; a &lt; 10, 1 \u2264 n \u2264 100000, show how to compute the value of 1 \u00d7 a + 2 \u00d7 a^2 + 3 \u00d7 a^3 + . . . + n \u00d7 a^n efficiently, i.e. in O(log n)!\" class=\"read-more\" href=\"https:\/\/jassweb.com\/solved\/solved-given-1-a-10-1-%e2%89%a4-n-%e2%89%a4-100000-show-how-to-compute-the-value-of-1-x-a-2-x-a2-3-x-a3-n-x-an-efficiently-i-e-in-olog-n\/\" aria-label=\"More on [Solved] Given 1 &lt; a &lt; 10, 1 \u2264 n \u2264 100000, show how to compute the value of 1 \u00d7 a + 2 \u00d7 a^2 + 3 \u00d7 a^3 + . . . + n \u00d7 a^n efficiently, i.e. in O(log n)!\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[320],"tags":[457,357],"class_list":["post-7937","post","type-post","status-publish","format-standard","hentry","category-solved","tag-algorithm","tag-math"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>[Solved] Given 1 &lt; a &lt; 10, 1 \u2264 n \u2264 100000, show how to compute the value of 1 \u00d7 a + 2 \u00d7 a^2 + 3 \u00d7 a^3 + . . . + n \u00d7 a^n efficiently, i.e. in O(log n)! - JassWeb<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/jassweb.com\/solved\/solved-given-1-a-10-1-\u2264-n-\u2264-100000-show-how-to-compute-the-value-of-1-x-a-2-x-a2-3-x-a3-n-x-an-efficiently-i-e-in-olog-n\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"[Solved] Given 1 &lt; a &lt; 10, 1 \u2264 n \u2264 100000, show how to compute the value of 1 \u00d7 a + 2 \u00d7 a^2 + 3 \u00d7 a^3 + . . . + n \u00d7 a^n efficiently, i.e. in O(log n)! - JassWeb\" \/>\n<meta property=\"og:description\" content=\"[ad_1] If this is homework I guess your professor explained you how to compute Fibonnacci numbers and expect you to use the same trick. 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